ON CONVOLUTIONS OF HARMONIC UNIVALENT MAPPINGS CONVEX IN THE DIRECTION OF THE REAL AXIS

Zhi-Gang Wang; Zhi-Hong Liu; Ying-Chun Li

doi:10.11948/2016012

2016 Volume 6 Issue 1

Article Contents

Article navigation > Journal of Applied Analysis & Computation > 2016 > 6(1): 145-155

Next Article Previous Article

Zhi-Gang Wang, Zhi-Hong Liu, Ying-Chun Li. ON CONVOLUTIONS OF HARMONIC UNIVALENT MAPPINGS CONVEX IN THE DIRECTION OF THE REAL AXIS[J]. Journal of Applied Analysis & Computation, 2016, 6(1): 145-155. doi: 10.11948/2016012

Citation:

Zhi-Gang Wang, Zhi-Hong Liu, Ying-Chun Li. ON CONVOLUTIONS OF HARMONIC UNIVALENT MAPPINGS CONVEX IN THE DIRECTION OF THE REAL AXIS[J]. Journal of Applied Analysis & Computation, 2016, 6(1): 145-155. doi: 10.11948/2016012

ON CONVOLUTIONS OF HARMONIC UNIVALENT MAPPINGS CONVEX IN THE DIRECTION OF THE REAL AXIS

1 School of Mathematics and Computing Science, Hunan First Normal University, Changsha 410205, Hunan, China;
2 School of Mathematics and Econometrics, Hunan University, Changsha 410082, Hunan, China;
3 College of Mathematics, Honghe University, Mengzi 661199, Yunnan, China

Fund Project:

Abstract

Abstract

In this paper, we show that convolutions of some planar harmonic mappings which convex in the direction of the real axis are also convex in the same direction. Furthermore, by means of the Mathematica software, we present an example to illuminate the main result.
- Harmonic univalent mapping /
- convolution /
- shear construction
MSC: 58E20;30C45

FullText(HTML)

References

[1]	A. Aleman and M. J. Martín, Convex harmonic mappings are not necessarily in h^1/2, Proc. Amer. Math. Soc., 143(2015), 755-763. Google Scholar
[2]	Z. Boyd, M. Dorff, M. Nowak, M. Romney and M. Woloszkiewicz, Univalency of convolutions of harmonic mappings, Appl. Math. Comput., 234(2014), 326-332. Google Scholar
[3]	J. Clunie and T. Sheil-Small, Harmonic univalent functions, Ann. Acad. Sci. Fenn. Ser. A. I Math., 9(1984), 3-25. Google Scholar
[4]	X.-D. Chen and A.-N. Fang, Estimation of hyperbolically partial derivatives of ρ-harmonic quasiconformal mappings and its applications, Complex Var. Elliptic Equ., 60(2015), 875-892. Google Scholar
[5]	M. Chuaqui, H. Hamada, R. Hernández and G. Kohr, Pluriharmonic mappings and linearly connected domains in Cn, Israel J. Math., 200(2014), 489-506. Google Scholar
[6]	A. Cohn, Uber die Anzahl der Wurzeln einer algebraischen Gleichung in einem Kreise, Math. Z., 14(1922), 110-148. Google Scholar
[7]	E. Deniz and H. Orhan, Lowner chains and univalence criteria related with Ruscheweyh and Salagean derivatives, J. Appl. Anal. Comput., 5(2015), 465-478. Google Scholar
[8]	M. Dorff, Convolutions of planar harmonic convex mappings, Complex Var. Theory Appl., 45(2001), 263-271. Google Scholar
[9]	M. Dorff, M. Nowak and M. Woloszkiewicz, Convolutions of harmonic convex mappings, Complex Var. Elliptic Equ., 57(2012), 489-503. Google Scholar
[10]	P. Duren, Univalent functions, Springer-Verlag, New York, 1983. Google Scholar
[11]	P. Duren, Harmonic mappings in the plane, Cambridge University Press, Cambridge, 2004. Google Scholar
[12]	M. Goodloe, Hadamard products of convex harmonic mappings, Complex Var. Theory Appl., 47(2002), 81-92. Google Scholar
[13]	X.-Z. Huang, Harmonic quasiconformal mappings on the upper half-plane, Complex Var. Elliptic Equ., 58(2013), 1005-1011. Google Scholar
[14]	Y.-P. Jiang, A. Rasila and Y. Sun, A note on convexity of convolutions of harmonic mappings, Bull. Korean Math. Soc., in press, 2015. Google Scholar
[15]	D. Kalaj, Quasiconformal harmonic mappings between Dini-smooth Jordan domains, Pacific J. Math., 276(2015), 213-228. Google Scholar
[16]	R. Kumar, M. Dorff, S. Gupta and S. Singh, Convolution properties of some harmonic mappings in the right-half plane, arXiv:1304.6167v1, 2013. Google Scholar
[17]	R. Kumar, S. Gupta, S. Singh and M. Dorff, An application of Cohn's rule to convolutions of univalent harmonic mappings, arXiv:1306.5375v1, 2013. Google Scholar
[18]	R. Kumar, S. Gupta, S. Singh and M. Dorff, On harmonic convolutions involving a vertical strip mapping, Bull. Korean Math. Soc., 52(2015), 105-123. Google Scholar
[19]	L.-L. Li and S. Ponnusamy, Solution to an open problem on convolutions of harmonic mappings, Complex Var. Elliptic Equ., 58(2013), 1647-1653. Google Scholar
[20]	L.-L. Li and S. Ponnusamy, Convolutions of harmonic mappings convex in one direction, Complex Anal. Oper. Theory, 9(2015), 183-199. Google Scholar
[21]	L.-L. Li and S. Ponnusamy, Sections of stable harmonic convex functions, Nonlinear Anal., 123-124(2015), 178-190. Google Scholar
[22]	W.-J. Li and Q.-H. Xu, On the properties of a certain subclass of close-toconvex functions, J. Appl. Anal. Comput., 5(2015), 581-588. Google Scholar
[23]	Z.-H. Liu and Y.-C. Li, The properties of a new subclass of harmonic univalent mappings, Abstr. Appl. Anal., Article ID 794108, 2013. Google Scholar
[24]	S. Muir, Harmonic mappings convex in one or every direction, Comput. Methods Funct. Theory, 12(2012), 221-239. Google Scholar
[25]	S. Nagpal and V. Ravichandran, A subclass of close-to-convex harmonic mappings, Complex Var. Elliptic Equ., 59(2014), 204-216. Google Scholar
[26]	S. Nagpal and V. Ravichandran, Convolution properties of the harmonic Koebe function and its connection with 2-starlike mappings, Complex Var. Elliptic Equ., 60(2015), 191-210. Google Scholar
[27]	C. Pommenrenke, On starlike and close-to-convex functions, Proc. London Math. Soc., 13(1963), 290-304. Google Scholar
[28]	S. Ruscheweyh and T. Sheil-Small, Hadamard products of schlicht functions and the Polya-Schoenberg conjecture, Comment. Math. Helv., 48(1973), 119-135. Google Scholar
[29]	D. Partyka and K. Sakan, Quasiconformal and Lipschitz harmonic mappings of the unit disk onto bounded convex domains, Ann. Acad. Sci. Fenn. Math., 39(2014), 811-830. Google Scholar
[30]	M. San and H. Irmak, Some novel applications of certain higher order ordinary complex differential equation to normalized analytic functions, J. Appl. Anal. Comput., 5(2015), 479-484. Google Scholar
[31]	Z.-G. Wang, Z.-H. Liu and Y.-C. Li, On the linear combinations of harmonic univalent mappings, J. Math. Anal. Appl., 400(2013), 452-459. Google Scholar
[32]	Z.-G. Wang, L. Shi and Y.-P. Jiang, Construction of harmonic univalent mappings convex in one direction, Sci. Sin. Math., 44(2014), 139-150. Google Scholar